The Moyal bracket and the dispersionless limit of the KP hierarchy

TL;DR

This paper introduces a novel Lax equation for the KP hierarchy using the Moyal bracket, simplifying the dispersionless limit and avoiding pseudo-differential operators, thus providing a more direct approach to the dispersionless KP hierarchy.

Contribution

It presents a new Lax formulation for the KP hierarchy based on the Moyal bracket, offering a simpler and more direct method for studying the dispersionless limit.

Findings

The new Lax equation avoids pseudo-differential operators.

The Moyal bracket simplifies the dispersionless limit.

The approach aligns with the study of dispersionless KP hierarchy.

Abstract

A new Lax equation is introduced for the KP hierarchy which avoids the use of pseudo-differential operators, as used in the Sato approach. This Lax equation is closer to that used in the study of the dispersionless KP hierarchy, and is obtained by replacing the Poisson bracket with the Moyal bracket. The dispersionless limit, underwhich the Moyal bracket collapses to the Poisson bracket, is particularly simple.